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Resolvent estimates for the Stokes operator on an infinite layer. (English) Zbl 1212.35343
Summary: In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer \(\Omega =\mathbb R^{n-1} \times (-1,1)\), \(n\geq 2\), in \(L^q\) Sobolev spaces, \(1<q<\infty \), with nonslip boundary condition \(u| _{\partial \Omega }=0\). The unique solvability is proved for every \(\lambda \in \mathbb C\setminus (-\infty ,-\pi ^2/4]\), where \(-\frac {\pi ^2}{4}\) is the least upper bound of the spectrum of Dirichlet realization of the Laplacian and the Stokes operator in \(\Omega \). Moreover, we provide uniform estimates of the solutions for large spectral parameter \(\lambda \) as well as \(\lambda \) close to \(-\frac {\pi ^2}{4}\). Because of the special geometry of the domain, a partial Fourier transformation is used to calculate the solution explicitly. Then Fourier multiplier theorems are used to estimate the solution operator.

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
Stokes operator